"Signal and System Study Notes" - Time Domain and Frequency Domain Characteristics of Signals and Systems (1)

Note: This blog is based on the second edition of Oppenheim's "Signals and Systems", mainly for the review and deepening of their own learning.

1. Modulus and phase representation of the Borrie transform

1. In general, the Borrie transform is complex-valued and can be represented by its real and imaginary parts, or its modulus and phase.

1) The modulus-phase representation of the continuous-time Borier transform X(jew) is

2) The modulus-phase representation of the discrete-time Borier transform X(jew) is

2. From the comprehensive formula of Bollier transform, X(jw) itself can be regarded as a decomposition of signal x(t), that is, the signal x(t) is decomposed into the sum of complex exponents of different frequencies. It can be regarded as the energy spectral density of x(t); that is to say, it can be regarded as the energy occupied by an infinitely small frequency band in the signal x(t) located in the frequency range from w to w+dw. Therefore, the fundamental frequency content of a signal is described by the modulus , and the relative amplitude information of each complex exponential signal that constitutes x(t) is given by the leaf.

3. The phase angle does not affect the magnitude of each frequency component, but provides relative phase information about these complex exponential signals. The phase relationship represented by it has a significant effect on the essential properties of the signal x(t) and therefore generally contains a lot of information about the signal. In particular, depending on what the phase function is, even if the modulo function remains the same, you can get signals that look very different. In general, changes in the phase function of x(t) lead to changes in the time-domain characteristics of the signal x(t). In some cases, phase eczema may be important, and in others, it may not be.

2. Modulus and phase representation of the frequency response of a linear time-invariant system

1. The effect of a linear four-invariant system on the input is to change the complex amplitude of each frequency component in the signal. Using the mode-phase representation, the nature of this effect can be explained in more detail. Specifically, in the continuous time case

and

There is a completely similar relationship in the discrete-time case.

1) It can be seen from the above formula that the effect of an emerging time-invariant system on the characteristics of the input Borie transform model is to multiply it by the mode of the system frequency response, which is generally called the gain of the system.

2) It can be seen from you that the phase input by the linear time-invariant system is transformed into a phase added on its basis , so it becomes a phase shift. The phase shift of the system can change the relative phase relationship between the components in the input signal, so that even if the gain of the system is constant for all frequencies, it is possible to thrive on this film and phase with large transformations.

1), linear and nonlinear phase

1. When the phase shift is a linear function of w, there is a very direct explanation for the effect of the phase shift in more than ten kinds. Consider the frequency response as

The continuous-time linear time-invariant system of , which has unity gain and fragrance shift, i.e.

The output that a system with such frequency response characteristics thrives on is the time shift of the input, i.e.

In the discrete-time case, when the slope of the linear phase is an integer, the effect is similar to that of love in the continuous-time case.

2. Although the transformation of a signal generated by linear phase shift is very simple and easy to understand and imagine, if the input signal received is a phase shift of a nonlinear function of w, then the complex frequencies of different frequencies in the input The exponential components will all be shifted in some way so that they are transformed above their phase. When the West Zhejiang complex indices are added together again, you get a signal that looks very different from the input signal.

2), group delay

1. A system with linear phase characteristics has the meaning of a special single piece, which is time shift. In fact, the phase characteristic is the magnitude of the time shift. That is to say, in the case of continuous time, if , then the time shift given by the system is -t0, or equivalently, the delay t0. Similarly, in the case of discrete time, it corresponds to a delay of n0.

2. The concept of time delay can be naturally extended directly to the case of nonlinear phase characteristics. Group delay is an example of this. The group delay at each frequency is equal to the assignment of the phase characteristic frequency at that frequency, that is, the group delay is defined as

The concept of group delay can be directly applied to discrete-time systems.

3), logarithmic analog and phase diagram

1. In a continuous-time system, if the modulus of the Borier transform is displayed on a logarithmic magnitude scale, then the following formula

There will be an additive relationship, that is

There are exactly the same expressions in the discrete-time case.

2. If there is a logarithmic and phase diagram of the input Longlie transform and a linear time-invariant system frequency response, then the output Bollier transform can add the logarithmic diagrams of the two, and the phase diagram be obtained by adding. Tong Yan, since the frequency response of the cascade of a linear time-invariant system is the product of the brother's frequency response, the logarithmic and phase diagrams of the total frequency response of a cascaded system can be added separately. and ask for. In addition, displaying the modulus of the Borier transform on a logarithmic scale can also reveal details on a plastic-framed dynamic map.

3. The logarithmic scale generally used is in units, called decibels. For continuous-time systems, it is also common and useful to use logarithmic frequency coordinates. and graphs for are called Bode graphs. In the discrete-time case, the Bollier transform and the magnitude of the frequency response are often also expressed in dB, for the same reasons as in the internal-time case. However, logarithmic frequency coordinates are generally not used in the discrete-time case, since the frequency range to be considered is always limited. And the point (leaf ie linear asymptote) that a differential equation has does not apply to a differential equation.

3. Time Domain Characteristics of Ideal Frequency Selective Filter

1. A continuous-time ideal low-pass filter has a frequency response of the form

Likewise, the frequency response of a discrete-time ideal low-pass filter should be

An ideal low-pass filter has frequency selectivity of notation. That is, they pass all frequencies below the cutoff frequency wc (including in wc) without attenuation, while completely blocking all frequencies within the stopband (ie above wc). Here, these filters have zero-phase characteristics, so they do not introduce phase distortion. Even if the mode of the signal spectrum is not changed by the system, nonlinear phase characteristics can cause large changes in the time-domain characteristics of a signal.

2. The unit impulse response of a continuous-time ideal low-pass filter is

Similarly, the unit impulse response corresponding to the discrete-time ideal repeated breaking is

If a linear phase characteristic is added to either of the continuous and discrete-time ideal frequency responses, then the unit impulse response is simply delayed by an amount equal to a replica of the slope of the phase characteristic.

3. The unit step responses s(t) and s[n] of continuous-time and discrete-time ideal low-pass filters are shown in the following figure

As can be seen in both cases, the step response exhibits several properties that may be undesirable. In particular, these filters have step responses that are larger than their final steady state values ​​and exhibit an oscillatory behavior called ringing. In addition, the step response is the integral or summation of the unit impulse response, i.e.

Since the unit impulse response of an ideal filter has a main lobe extending from -π/wc to +π/wc, the step response changes significantly in value during this time interval. That is, the so-called rise time of the step response is also inversely proportional to the bandwidth of the correlation filter.

4. Discussion on the time-domain and frequency-domain characteristics of non-ideal filters

1. Inadequacy of ideal filters

1), may not always be required in practice.

2), the step response problem of the ideal low-pass filter. In both required time and discrete time cases, the step response gradually tends to a constant value equal to the step value. However, overshoot (overshoot) and oscillations are present near the trip point. In some cases of love, this time-domain characteristic is not desirable.

3), the ideal filter is impossible to achieve.

2. The practical significance of non-ideal filters

1), allow some flexibility in the copper band and stop band characteristics of the filter, and allow a gradual transition characteristic between the copper band and the stop band relative to the steep transition band of the ideal filter .

2) In addition to the silent characteristics in the frequency domain, in some cases, the requirements of the phase characteristics are also very important. In particular, a linear approach to the phase of the phenomenon within a repass band is often desirable. In order to control the appetite characteristics, the index requirements are generally placed on the step response of a filter.

3. For non-ideal low-pass filters, it can be seen that there may be an intentional compromise between the width of the transition band (frequency characteristics) and the settling time of the step response (time-domain characteristics). The trade-off between filter time-domain and frequency-domain characteristics, and consideration of issues such as the trade-off between complexity and cost, becomes a core area of ​​filter design.